Homomorphisms of a modular lattice

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Abstract

This thesis is an algebraic study of the irreducible ideals of a modular lattice and their application to the characterization of the homomorphisms or congruence relations of the lattice.

First, an arithmetic characterization of modularity is given in terms of the irreducible ideals of the lattice. This is a new structure result for modular lattices which, since it characterizes general modular lattices, is more fundamental in the structure of modular lattices than the Kurosh-Ore Theorem.

Second, through the arithmetic characterization developed, subsets of the irreducible ideals are used to define congruence relations on the lattice and its lattice of ideals. It is then shown that every congruence relation on a modular lattice can be so characterized.

In conclusion, a generalization of the theorem that the congruence relations of a finite dimensional modular lattice form a Boolean algebra is given by proving that the congruence relations on the lattice of ideals of a modular lattice form a Boolean algebra if and only if the lattice is finite dimensional.